A Binary Search Tree (abbreviated as BST) is a tree structure comprised of nodes that maintain a specific order among their keys. Given a single node $x$ and two BSTs $L$ and $R$, in which $l.key < x.key < r.key$ for all nodes $l \in L$ and $r \in R$, the objective is to design a $Join(L,x,R)$ function that integrates the given input into a unified BST $T$ with the nodes $L \cup \{x\} \cup R$. Be aware that AVL and Red-Black trees are specific types of BSTs. Given the constraint that $T$, $L$, and $R$ must share the same type, they are required to be either all BSTs, all AVL trees, or all Red-Black trees.

The $Join(L,x,R)$ function is implemented with the steps: (1) selecting a subtree $T'$ from either $L$ or $R$, where a subtree is defined as potentially being empty, a portion, or the entire tree, (2) detaching $T'$, (3) attaching the other tree (which does not contain $T'$) under $x$, (4) re-attaching the tree rooted at $x$ to the former parent of $T'$. The modified tree is then returned as $T$. If $T'$ has no former parent, the tree rooted at $x$ is simply returned as $T$. The following figure demonstrates an example result of $Join(L_1,x_1,R_1)$, where the letters $\alpha, \beta, \gamma, \delta,$ and $\eta$ denote arbitrary subtrees.

Please note that almost all steps, except for (1), have been finalized — there could be multiple ways to choose $T'$ in (1). The tree returned from $Join(L,x,R)$ may not be unique. Additionally, unlike AVL or Red-Black trees, $Join(L,x,R)$ does not perform re-coloring and re-balancing. Consider the following arguments passed into $Join(L_2,x_2,R_2)$. All nil nodes (considered as leaves in red-black trees) are omitted for brevity.

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